# An Alternative Measure of Angles 1 Name

Name CHAPTER 1 Date An Alternative Measure of Angles Use after Lesson 1.4 When you measured angles in the past, you used the unit of degrees. In this lesson we’ll learn about another unit used to measure angles, the radian. When measuring an angle in radians, you express the measurement in terms of pi (:). The radian measure is considered to be a real number so it is not necessary to also write the term “radians” when writing a measure in radians. If the measure of an angle does not have a degree symbol after it, it is assumed to be in radians. The unit circle is a circle with radius 1. Compare the two unit circles below. rad 2 90° 2 rad 3 rad 2 270° Pre–AP Copymasters 0 rad A complete revolution of the circle is measured to be 360° or 2: radians. 2: is the radian measurement while 360° is the degree measurement. When using your calculator to perform calculations with radian angle measurements, be sure to change the mode first. Otherwise, your calculations will not be correct. If you enter 2: in “degree” mode, your calculator will register the entry to be approximately 6.28°. In “radian,” mode your entry is equal to 360°. The Greek symbols theta (u) and phi (f) are commonly used to represent angle measurements in mathematics and physics. EXAMPLE 1 Convert from degrees to radians Convert 90° to radians. Solution: Looking at the unit circles above, you can see that 90° corresponds to a radian : measurement of } radians. N 2 EXAMPLE 2 Convert from radians to degrees 3: Convert } radians to degrees. 2 Solution: 3: Looking at the unit circles, you can see that } radians corresponds to 270°. N 2 EXAMPLE 3 Convert from degrees to radians Convert 27° to radians. 312 Geometry Best Practices Toolkit Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 0° 360° 180° Name CHAPTER 1 Date An Alternative Measure of Angles continued Use after Lesson 1.4 Solution: We know that : radians 5 180°. If we express this as a 1:1 ratio, we can use it to convert the angle measurement. : rad 3: rad 5} 5 0.15: rad 27° + } 180° 20 Check: 15: 27° is between 0° and 90° on the unit circle, so } radians should be between 0 and 100 : 2 } radians. N EXAMPLE 4 Convert from radians to degrees 3: Convert } radians to degrees. 4 Solution: Use the inverse of the ratio used in Example 3 to convert the measurement. 3: rad 4 180° : rad } + } 5 135° Check: 3: : } radians is between } radians and : radians on the unit circle, so 135° should 4 2 be between 90° and 180°. N Practice Convert the degree measurements to radians 2. 9° 3. 30° 4. 45° 5. 60° 6. 18° 7. 36° 8. 72° 9. 144° 10. 288° 11. 54° 12. 126° 13. 216° 14. 720° 15. 540° 16. 2160° 17. 1980° Convert the radian measurements to degrees : : : 18. } 19. } 20. } 6 3 4 5: 4: 4: 22. } 23. } 24. } 9 15 9 5: 11: : 26. } 27. } 28. } 9 3 15 30. 8: 31. 4: 32. 10: : 21. } 9 7: 25. } 12 35: 29. } 18 33. 3: Pre–AP Copymasters Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 1. 1° is how many radians? Measure the angle indicated and convert it to radians 34. 35. u u Geometry Best Practices Toolkit 313

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